Absolute Values and Rational Numbers

Key Concepts

• Describe quantities using absolute value.
• Find absolute value.
• Interpret absolute value.

Introduction: 

Absolute value: 

The absolute value of a number on the number line is defined to be the distance that point is from zero. Since, the absolute value is defined in terms of distance, it can never be negative. To evaluate the absolute value of a number, state the distance it is from zero.  

The absolute value of -15 is 15″ is written like this: |-15| = 15. 

Since, the absolute value of a number is that number’s distance from 0, the absolute value of a number is always 0 or positive. It is never negative.  

2.3.1 Describe quantities using an absolute value 

Example 1: 

A stock’s price gained 3% in April and 5% in May, and then lost 4% in June and 1% in July. During which month did the stock’s price change the most?  

Solution: 

Gain is represented as a positive number and loss is represented as a negative number. 

The absolute values of the changes in the company’s stock price each month are, 

Example 2: 

Melissa starts on the 10th floor of a building and takes the elevator 3 floors down. Then she takes the elevator up to 4 floors, and then down to another 5 floors. Write the absolute value of the greatest change in floors that Melissa made.  

Solution: 

Find the absolute values of the changes in floors. 

The absolute values of the changes in the floors Melissa made are, 

2.3.2 Find absolute value 

Example 3: 

Find the absolute value of each of these. 

1. |-4| b.     |-1| c.    |3|  

Solution: 

1. Absolute value of |-4| 

2. Absolute value of |-1| 

3. Absolute value of |3| 

2.3.3 Interpret absolute value 

Example 4: 

Marie’s account balance is –$45.62. Jason’s account balance is –$42.55. Which balance represents the greater number? Which balance represents the lesser amount owed?  

Solution: 

Because –45.62 is farthest to the left on the number line than –42.55.  –45.62 is the least number and the greater number is –42.55. 

The absolute value of each balance describes the size of each debt, or the amount of money owed. 

Example 5: 

The table shows the daily change in high temperature for several days. Explain how you can order the days from the least to the greatest amount of temperature change.   

Solution: 

Find the absolute value of each change. 

Order the days by the absolute values, 

1 < 2 < 3 < 4  

Exercise:

1. How do you know that |8| and |-8|are the same distance from 0? Do they have the same absolute value? Explain.
2. Name two numbers that are not located the same distance from 0. What are their absolute values?
3. Which pair of numbers are located the same distance from 0 on the number line?
   A. 5 and – 4              B. 0 and 1               C. – 3 and 3            D. – 2 and – 4
4. Evaluate the absolute value of each of the following numbers:
   a. 4: ____                           b. – 8: ____               c. 2: ____                  d. -1: ____                e. 0: ____
5. Write two numerals that have an absolute value of 6 ______ and _______
6. Write two numerals that have an absolute value of 19 ______ and ______
7. Write all numerals that have an absolute value of zero _______
8. On the number line, plot and label all the numbers with an absolute value of 3/2.
9. The temperature at dawn is 6^oC away from 0. Select all the temperatures that are possible.
   a. –12^oC
   b. –6^oC
   c. 0^oC
   d. 6^oC
   e. 12^oC
10. Of the three elevations, –2 feet, –12 feet, and 30 feet, which represents the least number? Which represents the farthest distance from sea level?

Concept Map: 

What have we learned:

• Describe quantities using absolute value in the given scenario.
• Find absolute value using a number line.
• Interpret absolute value in the given scenario.